Robust target identification

ABSTRACT

A target estimator that properly conditions measurement variates in the case of a series of sensor measurements collected against a target, a system model that captures visible and hidden stochastic information including but not limited to target state, target identity, and sensor measurements and that marginalizes measurement failure and a dynamic mixed quadrature expression facilitating real-time implementation of the estimator are presented.

BACKGROUND

The present invention relates to a system and method for robust targetidentification using a Bayes optimal estimation approach.

It is often desirable that sensing systems be able to exploit the datathey collect for interesting and relevant information. A task that oftenfalls on human analysts is the examination of sensor data for targets(e.g. vehicles, buildings, people) of interest and the labeling of suchobjects in a manner aligned with the collection objective (e.g.friend/foe, seen before/new, authorized/unauthorized, etc.). Targetdetection and identification is a fundamental problem in manyapplications, such as in hyperspectral imaging, computer-aideddiagnosis, geophysics, Raman spectroscopy and flying objectidentification. Under limited conditions, this task has been automatedby machine learning algorithms. Such algorithms are typically trainedwith examples drawn from the sensor data they are meant to later processautonomously.

The problem is fundamentally one of crafting a decision rule R:

→

that maps evidence (E∈

) to some element of the set of hypotheses (

),

-   -   H_(i): E arises from T_(i) (target i)    -   H_(o): E does not sufficiently support any target

Decision methods that provide a minimum decision error rate requireknowledge of the posterior distribution of target type (p(T|E)).

In many instances, the physical situation from which the problem arisesis expressed in a state variable representation. The internal statevariables are the smallest possible subset of system variables that canrepresent the entire state of the system at any given time. The statevariable representation can be expressed ass _(t) =A(t)s+W

Where s is the vector of state variables,

-   -   s_(t) is the time derivative of s,    -   A(t) is the state transition matrix, and    -   W is a noise vector.

Some of the states can be related to measurements while other states arenot related to measurements due to sensor limitations, excessive noiseand other factors. The states that are not related to the measurementsare often referred to as “hidden variables” or “hidden states.”

It is generally the case that the dimensionality of

is high and so a common approach is to use expert knowledge of thesystem to develop a projection to a lower dimensional measurement spaceh:

→

₁×

₂ . . . ×

_(F), where we presume F distinct measurement variates made from E. Thisapproach has the distinct advantage of reducing training sample sparsityand allowing the designer to exploit ‘features’ with desirableproperties known a-priori. This approach has the unfortunate consequenceof complicating the calculation of p(T|E). In particular, unlessproperly conditioned, the measurement variates are correlated and maynot be simply combined.

In many conventional cases, the hidden states are replaced by estimatedvalues.

In real systems, the input data can be intermittently and unpredictablydegraded. Pre-screening is a common but sub-optimal solution.

There is a need for target identification methods and systems that allowfor optimal estimation and marginalization of measurement failure.

There is a need for a target estimator that properly conditionsmeasurement variates in the case of a series of sensor measurementscollected against a target.

There is also a need for a system model that captures visible and hiddenstochastic information including but not limited to target state, targetidentity, and sensor measurements.

There is a further need for a dynamic mixed quadrature expressionfacilitating real-time implementation of the estimator.

There is still a further need for target identification methods andsystems that take into account data degradation.

BRIEF SUMMARY

A target estimator that properly conditions measurement variates in thecase of a series of sensor measurements collected against a target, asystem model that captures visible and hidden stochastic informationincluding but not limited to target state, target identity, and sensormeasurements and that marginalizes measurement failure and a dynamicmixed quadrature expression facilitating real-time implementation of theestimator are presented herinbelow.

In one or more embodiments, the method for target identification ofthese teachings includes receiving, at one or more processors, a numberof measurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), each one ofthe number of measurements, each one a number of target types (T), andeach one of one or more hidden states, each hidden state (x_(k)) beingcharacterized at the predetermined time, being correlated to oneanother, a number of measurement quality values (q_(k)), one measurementquality value for each of the number of measurements, the a number ofmeasurement quality values being characterized at the predeterminedtime, providing, using the one or more processors, a first conditionalprobability distribution, a conditional probability of a target typegiven a number of measurements, defined inductively by

${{p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}},{And}$$\begin{matrix}{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}}\end{matrix}$where p(z_(k,i)|T,x_(k)) is a conditional probability of an ithmeasurement at a kth instance giving a target type T and hidden statesat the kth instance, andp(T|z_(1, 2, . . . k−1)) is a conditional probability distributionfunction of a target type, T, given the plurality of measurements at thek−1^(th) time, z_(1, . . . k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0;p(z_(k), x_(k), q_(k)|T, z_(1, 2, . . . , k−1)) is a conditionalprobability distribution function of a measurement at the kth instance,hidden states at the kth instance, measurement quality at the kthinstance given a target type and the plurality of measurements at thek−1^(th) time, z_(1, . . . k−1);p(z_(k)|T, x_(k), q_(k)) is a conditional probability distributionfunction of the measurement at the kth instance given a target type, thehidden states at the kth instance and the measurement quality at the kthinstance; andp(x_(k), q_(k)|T, z_(1, 2, . . . , k−1)) is a conditional probabilitydistribution function of the measurement at the kth instance and themeasurement quality at the kth instance given a target type and theplurality of measurements at the k−1^(th) time, z_(1, . . . k−1) andobtaining an estimate of the target type from the first conditionalprobability.

In one or more other embodiments, the method for target identificationof these teachings includes receiving, at one or more processors, anumber of measurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), each one ofthe number of measurements, each one a number of target types (T), andeach one of one or more hidden states, each hidden state (x_(k)) beingcharacterized at the predetermined time, being correlated to oneanother, a number of measurement quality values (q_(k)), one measurementquality value for each of the number of measurements, the a number ofmeasurement quality values being characterized at the predeterminedtime, providing, using the one or more processors, a first conditionalprobability distribution, a conditional probability of a target typegiven a number of measurements, defined inductively by

${{p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}},{And}$$\begin{matrix}{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}}\end{matrix}$where p(z_(k,i)|T,x_(k)) is a conditional probability of an ithmeasurement at a kth instance giving a target type T and hidden statesat the kth instance, andp(T|z_(1, 2, . . . k−1)) is a conditional probability distributionfunction of a target type, T, given the plurality of measurements at thek−1^(th) time, z_(1, . . . k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0;p(z_(k), x_(k), q_(k)|T, z_(1, 2, . . . , k−1)) is a conditionalprobability distribution function of a measurement at the kth instance,hidden states at the kth instance, measurement quality at the kthinstance given a target type and the plurality of measurements at thek−1^(th) time, z_(1, . . . k−1);p(z_(k)|T, x_(k), q_(k)) is a conditional probability distributionfunction of the measurement at the kth instance given a target type, thehidden states at the kth instance and the measurement quality at the kthinstance; andp(x_(k), q_(k)|T, z_(1, 2, . . . , k−1)) is a conditional probabilitydistribution function of the measurement at the kth instance and themeasurement quality at the kth instance given a target type and theplurality of measurements at the k−1^(th) time, z_(1, . . . k−1);and obtaining an estimate of the target type from the first probabilitydistribution.

In one or more embodiments, the target identification system of theseteachings includes a memory for storing data, the memory having a datastructure stored in the memory, the data structure including a number ofmeasurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), a number oftarget types (T), each one of the number of measurements, each one thenumber of target type and each one of one or more hidden states, eachhidden state (x_(k)) being characterized at the predetermined time,being correlated to one another, and one or more processors operativelyconnected to the memory for storing data; the one or more processorsbeing configured to implement one embodiment of the method of theseteachings for target identification.

A number of other embodiments are also disclosed.

For a better understanding of the present teachings, together with otherand further objects thereof, reference is made to the accompanyingdrawings and detailed description and its scope will be pointed out inthe appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is shows one embodiment of the stochastic system model of theseteachings;

FIG. 2 is a block diagram representation depicting operation of oneembodiment of the system of these teachings;

FIG. 2a is a block diagram representation of one embodiment of thesystem of these teachings;

FIGS. 3a-3c are embodiments of the stochastic system model of theseteachings;

FIG. 4 is a flowchart representation of the adaptive quadrature methodof these teachings;

FIG. 5 is a flowchart representation of one embodiment of the method ofthese teachings; and

FIG. 6 represents a comparison of results of the method of theseteachings and conventional methods.

DETAILED DESCRIPTION

The following detailed description is of the best currently contemplatedmodes of carrying out these teachings. The description is not to betaken in a limiting sense, but is made merely for the purpose ofillustrating the general principles of these teachings, since the scopeof these teachings is best defined by the appended claims.

The present teachings will be more completely understood through thefollowing description, which should be read in conjunction with thedrawings. In this description, like numbers refer to similar elementswithin various embodiments of the present disclosure. Within thisdescription, the claims will be explained with respect to embodiments.The skilled artisan will readily appreciate that the methods, apparatusand systems described herein are merely exemplary and that variationscan be made without departing from the spirit and scope of thedisclosure. As used herein, the singular forms “a,” “an,” and “the”include the plural reference unless the context clearly dictatesotherwise.

Except where otherwise indicated, all numbers expressing quantities ofingredients, reaction conditions, and so forth used in the specificationand claims are to be understood as being modified in all instances bythe term “about.”

Before describing the present teachings in detail, certain terms aredefined herein for the sake of clarity.

“Fisher information,” as used herein, is a measure of the amount ofinformation that an observable random variable X carries about anunknown parameter θ upon which the probability of X depends.

“Quadrature,” as used herein, is an approximation of the definiteintegral of a function, usually stated as a weighted sum of functionvalues at specified points within the domain of integration.

Adaptive quadrature, as used herein, describes approximating a definiteintegral of a function by a sum of approximations of definite integralsover subintervals.

In one type of quadrature, the quadrature rule is constructed to yieldan exact result for polynomials of degree 2n−1 or less by a suitablechoice of the points x_(i) and weights w_(i) for i=1, . . . , n.

The order of a quadrature rule, or quadrature order, as used herein, isthe degree of the lowest degree polynomial that the rule does notintegrate exactly.

A knot, as used herein, is a point at which an approximation over oneinterval connects continuously to an approximation over anotherinterval.

A “whitening transformation,” as used herein, is a decorrelationtransformation that transforms a set of random variables having a knowncovariance matrix M into a set of new random variables whose covarianceis the identity matrix (meaning that they are uncorrelated and all havevariance).

“Kronecker product,” as used herein, denoted by

, is an operation on two matrices of arbitrary size resulting in a blockmatrix.

The “Khatri-Rao product,” as used herein, is defined asA*B=*A _(ij)

B _(ij))_(ij)in which the ij-th block is the m_(i)p_(i)×n_(j)q_(j) sized Kroneckerproduct of the corresponding blocks of A and B, assuming the number ofrow and column partitions of both matrices is equal.

The “Bayes classifier,” as used herein, is defined as

${C^{Bayes}(x)} = {\underset{r \in {\{{1,2,\ldots,K}\}}}{argmax}{{P\left( {Y = {\left. r \middle| X \right. = x}} \right)}.}}$

Although the invention has been described with respect to variousembodiments, it should be realized these teachings are also capable of awide variety of further and other embodiments within the spirit andscope of the appended claims.

A target estimator that properly conditions measurement variates in thecase of a series of sensor measurements collected against a target, asystem model that captures visible and hidden stochastic informationincluding but not limited to target state, target identity, and sensormeasurements and that marginalizes measurement failure and a dynamicmixed quadrature expression facilitating real-time implementation of theestimator are presented herinbelow.

The method and system of these teachings include a stochastic systemmodel, capturing visible and hidden stochastic information including butnot limited to target state, target identity, sensor measurements, andmeasurement quality and a Bayes' optimal estimator of target identityover multiple coherent sensor integration periods. In one instance, themethod and system of these teachings also include a dynamic mixedquadrature facilitating real-time implementation of the estimator.

The stochastic system model of these teachings, shown in FIG. 1,captures correlation between target type, target state, andmeasurements. Referring to FIG. 1,

represents and contains a hidden random variable,

represents and contains an observable random variable, and

represents the correlation between two random variables. In thestochastic system model shown in FIG. 1,

x_(l)={x_(l,1), x_(l,2), . . . , x_(l,M)}: all relevant time-dependenthidden states of target at look l∈{1, 2, . . . K}

z_(l)={z_(l,1), z_(l,2), . . . , z_(l,F)}: F target measurements at lookl

q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: F measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0

T: hidden target type.

This system model of these teachings can be differentiated from currenttarget identification (CID) that treat consecutive measurements asindependent and identically distributed (i.i.d.). This is because oversmall periods of time target state (of which measurements are afunction) is a correlated Markov process. This methodology provides anadditional benefit in that measurements exhibit statistical independencewhen conditioned on target state and type, that isp(z _(k,i) ,z _(k,j) |x _(k) ,T)=p(z _(k,i) |x _(k) ,T)p(z _(k,j) |x_(k) ,T),i≠j

The joint density of hidden variables includes K×F×M+1 variates, whichmust be marginalized to yield a target type posterior. For a typicalproblem this would imply hundreds or thousands of variables.

The Bayes Optimal Estimator of these teachings is disclosed hereinbelow.

The goal is an estimate of p(T|z_(1, 2, . . . k)) at each look k.

Inductively, starting from p(T|z_(1, 2, . . . , k−1)), then

${p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}$$\begin{matrix}{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{{qk} = 0},1}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\;{x_{k}.}}}}}}\end{matrix}$Alternatively, p(T|z_(1, 2, …k)) = ∫p(x_(k), q_(k), T|, z_(1, 2, …, k))d x_(k) × d q_(k)${p\left( {x_{k},q_{k},\left. T \middle| z_{1,2,\ldots,k} \right.} \right)} = \frac{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}{p\left( z_{k} \middle| z_{1,2,\ldots,{k - 1}} \right)}$now, p(x_(k), x_(k − 1), q_(k), q_(k − 1)|T, z_(1, 2, …, k − 1)) = p(x_(k), q_(k)|x_(k − 1), q_(k − 1), T, z_(1, 2, …, k − 1))p(x_(k − 1), q_(k − 1)|T, z_(1, 2, …, k − 1))p(x_(k), q_(k)|T, z_(1, 2, …, k − 1)) = ∫p(x_(k), q_(k)|x_(k − 1), q_(k − 1), T, z_(1, 2, …, k − 1))p(x_(k − 1), q_(k − 1)|T, z_(1, 2, …, k − 1))d x_(k − 1) × q_(k − 1)p(x_(k), q_(k)|T, z_(1, 2, …, k − 1)) = ∫p(x_(k)|x_(k − 1), T)p(q_(k), q_(k − 1), T)p(x_(k − 1), q_(k − 1)|T, z_(1, 2, …, k − 1))d x_(k − 1) × q_(k − 1)${thus},\begin{matrix}{{p\left( {z_{k},q_{k},\left. T \middle| z_{1,2,\ldots,k} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}\frac{{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\int{{p\left( {\left. x_{k} \middle| x_{k - 1} \right.,T} \right)}{p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1} \times q_{k - 1}}}}{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}}} \\{= {\underset{i = {1\ldots\; F}}{\Pi}{p\left( {\left. z_{k,i} \middle| T \right.,x_{k},q_{k,i}} \right)}\frac{{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\int{{p\left( {\left. x_{k} \middle| x_{k - 1} \right.,T} \right)}{p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1} \times q_{k - 1}}}}{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}}}\end{matrix}$and the target-conditional target state posterior is recursivelycalculated.

This formulation provides a framework for the fusion of features, whichare integrated in a manner which makes them independent of one-another.It also includes optimal treatment of the correlation between targettype, observed features, and time-dependent target state and measurementquality. An inductive formulation allows for online implementation,however each step of the recursion requires a number of integrationsover target state and measurement quality, which requires numericalfinesse for re al time implementation on modern microprocessors.

In order to start the induction, the initial joint probability can beestimated from measurements (see, for example, Mauricio Monsalve, Amethodology for estimating joint probability density functions, Apr. 15,2009, a copy of which is incorporated by reference herein in itsentirety and for all purposes) or can be assumed to be a known form,typically Gaussian, with estimated parameters.

In one or more embodiments, the method for target identification ofthese teachings includes receiving, at one or more processors, a numberof measurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), a number oftarget types (T), each one of the number of measurements, each one thenumber of target type and each one of one or more hidden states, eachhidden state (x_(k)) being characterized at the predetermined time,being correlated to one another, providing, using the one or moreprocessors, a first conditional probability distribution, a conditionalprobability of a target type given a number of measurements, definedinductively by

${p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}$${p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}} = {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}}$where p(z_(k,i)|T,x_(k)) is a conditional probability of an ithmeasurement at a kth instance giving a target type T and hidden statesat the kth instance, and p(T|z_(1, 2, . . . k−1)) is a conditionalprobability distribution f of a target type, T, given the plurality ofmeasurements at the k−1^(th) time, z_(1, . . . k−1), and obtaining anestimate of the target type from the first conditional probability.

In one embodiment, the estimate is obtained using a Bayes Classifier. Itshould be noted that other decision rules, such as a likelihood ratiotest and even decision rules under the Neyman Pearson formalism arewithin the scope of these teachings.

In one instance, wherein p(x_(k)|x_(k−1),T) is a multivariate Gaussiandistribution and wherein p(x_(k)|T, z_(k)) is another multivariateGaussian distribution, where p(x_(k)|x_(k−1),T) is a conditionalprobability distribution of hidden states at instance k given hiddenstates at instance k−1 and target type T. In one embodiment of thatinstance, transition between one hidden state at one instance and theone hidden state at another instance is given by a predetermined dynamicmodel (referred to as ƒ_(k)(x_(k−1), T)) and wherein an expectation ofp(x _(k) |T,z _(1,2) , . . . k−1)is given byp(x _(k) |T,z _(1,2, . . . , k−1))=N(x _(k);μ_(k,k−1) ^(x|T) ,P _(k,k−1)^(xx|T))μ_(k,k−1) ^(x|T)=∫ƒ_(k)(x _(k−1))N(x _(k);μ_(k−1,k−1) ^(x|T) ,P_(k−1,k−1) ^(xx|T))dx _(k−1)and a covariance matrix is given inductively byP _(k,k−1) ^(xx|T) =Q _(k)+∫ƒ_(k)(x _(k−1))ƒ_(k) ^(T)(x _(k−1))N(x_(k);μ_(k−1,k−1) ^(x|T) ,P _(k−1,k−1) ^(xx|T))dx _(k−1)−[μ_(k,k−1)^(x|T)]^(T)μ_(k,k−1) ^(x|T).

In the above embodiment, p(x_(k)|T, z_(1, 2, . . . k−1)) is amultivariate Gaussian distribution. In one instance, measurements are apredetermined function of target state and type (referred to ash_(k)(x_(k), T).

In one embodiment, the measurements z_(i) include one or more of a rangeto the target, a rate of change of a range to the target, anacceleration along a line of sight.

In one or more embodiments, the hidden states x_(i) include one or moreof target geometry, target attitude, target velocity.

In one or more embodiments, the target types include one or more oftypes of vehicles. It should be noted that these teachings are notlimited to only those embodiments.

Recognition of targets is a problem that is found in a large number oftechnical areas and the present teachings applicable to all of thosetechnical areas.

In one or more embodiments, the target identification system of theseteachings includes a memory for storing data, the memory having a datastructure stored in the memory, the data structure including a number ofmeasurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), a number oftarget types (T), each one of the number of measurements, each one thenumber of target type and each one of one or more hidden states, eachhidden state (x_(k)) being characterized at the predetermined time,being correlated to one another, and one or more processors operativelyconnected to the memory for storing data; the one or more processorsbeing configured to implement one embodiment of the method of theseteachings for target identification.

FIG. 2 shows operation 100 of one embodiment of the targetidentification system of these teachings. Referring to FIG. 2, a target106 is detected by a sensor 104 and the output of the sensor is storedin a memory 110 and provided to the target identification system 102. Inthe embodiment shown, the system is distributed and the sensor 104, thememory 110 and the target identification system 102 are connectedthrough a network. It should be noted that other embodiments in whichthe system is not distributed are also within the scope of theseteachings.

FIG. 2a shows one embodiment of the target identification system ofthese teachings. Referring to FIG. 2a , in the embodiment shown therein, the system has one or more processors 115, can receive input fromthe sensor via an input component 120 and the one or more processors 115are operatively connected to the memory 110 and to computer usable media125 that has computer readable code embodied therein, which, whenexecuted by the one or more processors 115, causes the one or moreprocessors 115 to perform an embodiment of the method for targetidentification of these teachings. The one or more processors 115 areoperatively connected to the memory 110 and to computer usable media 125by a connection component 130, which can be a single connectioncomponent, such as a computer bus, or a combined connection componentincluding a network, for distributed components, and another connectioncomponent for parts of the system that are physically connected.

The inductive formulation, although amenable to online implementation,requires a large number of integrations over the target state. Acomputational strategy that implements a recursive estimator in a mannerthat is more easily implemented in a real time system is presentedherein below.

In the first step of the computational strategy, a target type dependentdynamic model is used to capture the state evolution betweenmeasurements. The first step is shown schematically in the stochasticsystem model diagram shown in FIG. 3 a.

The state transition is considered as the sum of a deterministicfunction (ƒ_(k)) of the previous state and independent Gaussian noise(below the function N(⋅; u, P) denotes a multivariate Gaussian with meanu and covariance P),p(x _(k) |x _(k−1) ,T)=N(x _(k);ƒ_(k)(x _(k−1) ,T),Q _(k)).

The measurement quality is considered to be described by a Bernoullidistribution and the conditional probability of a measurement quality atthe kth instance given a measurement quietly at the (k−1)th instance andthe target type is given by

${p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)} = {{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)} = \left\{ {{\begin{matrix}{M_{k}q_{k - 1}} & {q_{k} = q_{k - 1}} \\{\left( {1 - M_{k}} \right)q_{k - 1}} & {q_{k} \neq q_{k - 1}}\end{matrix}{Then}\text{:}\mspace{14mu}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}} = {{\sum\limits_{q_{{{k - 1} = 0},1}}{\int{{p\left( {\left. x_{k} \middle| x_{k - 1} \right.,T} \right)}{p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}}}} = {\sum\limits_{q_{{{k - 1} = 0},1}}{\int{{N\left( {{x_{k};{f_{k}\left( {x_{k - 1},T} \right)}},Q_{k}} \right)}{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}\begin{matrix}{{E\left\{ {\left. x_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right\}} = {\int{x_{k}{\sum\limits_{q_{{k = 0},1}}{{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}} \\{= {\int{x_{k}{\sum\limits_{q_{{k = 0},1}}{\left\lbrack {\sum_{q_{{{k - 1} = 0},1}}{\int{{N\left( {{x_{k};{f_{k}\left( x_{k - 1} \right)}},Q_{k}} \right)}{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}} \right\rbrack d\; x_{k}}}}}} \\{= {\sum\limits_{q_{{k = 0},1}}{\sum\limits_{q_{{{k - 1} = 0},1}}{{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)}{\int{\left\lbrack {\int{x_{k}{N\left( {{x_{k};{f_{k}\left( x_{k - 1} \right)}},Q_{k}} \right)}d\; x_{k}}} \right\rbrack{p\left( {x_{k -},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}}}}} \\{= {\sum\limits_{q_{{k = 0},1}}{\sum\limits_{q_{{{k - 1} = 0},1}}{{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)}{\int{{f_{k}\left( x_{k - 1} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}}}}} \\{= {\sum\limits_{q_{{{k - 1} = 0},1}}{\int{{f_{k}\left( x_{k - 1} \right)}{p\left( {\left. x_{k - 1} \middle| q_{k - 1} \right.,T,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}}}\end{matrix}\begin{matrix}{{E\left\{ {\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right\}} = {\sum\limits_{q_{{k = 0},1}}{q_{k}{\int{\left\lbrack {\sum\limits_{q_{{{k - 1} = 0},1}}{\int{{N\left( {{x_{k};{f_{k}\left( x_{k - 1} \right)}},Q_{k}} \right)}{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}} \right\rbrack d\; x_{k}}}}}} \\{= {\int{\left\lbrack {\sum\limits_{q_{{{k - 1} = 0},1}}{\int{{N\left( {{x_{k};{f_{k}\left( x_{k - 1} \right)}},Q_{k}} \right)}{B\left( {{1;q_{k - 1}},M_{k}} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k - 1}}}} \right\rbrack d\; x_{k}}}} \\{= {\sum\limits_{q_{{{k - 1} = 0},1}}{{B\left( {{1;q_{k - 1}},M_{k}} \right)}{p\left( {x_{k - 1},\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d_{k - 1}}}} \\{= {{M_{k}{p\left( {{q_{k - 1} = \left. 1 \middle| T \right.},z_{1,2,\ldots,{k - 1}}} \right)}} = {M_{k}E\left\{ {\left. q_{k - 1} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right\}}}}\end{matrix}}}}}} \right.}$

If the target type-conditional state posterior from the previousmeasurement is Gaussian,p(x _(k−1) |T,z _(1,2, . . . , k−1))=N(x _(k);μ_(k−1,k−1) ^(x|T) ,P_(k−1,k−1) ^(xx|T))where μ_(m,n) ^(a|T) denotes E{a|T} at step m using evidence up to stepn and P_(m,n) ^(a)=E{aa^(T)} and it is considered thatp(x _(k−1) ,q _(k−1) |T,z _(1,2, . . . , k−1))=N(x _(k);μ_(k−1,k−1)^(x|q) ^(k) ,P _(k−1,k−1) ^(xx|q) ^(k) )B(q _(k);μ_(k−1,k−1) ^(q) ^(k),1)where μ_(m,n) ^(a) denotes E{a} at step m using evidence up to step nand P_(m,n) ^(a)=E{aa^(T)},Then,p(x _(k) ,q _(k) |T,z _(1,2, . . . , k−1))=N(x _(k);μ_(k,k−1) ^(x|q)^(k) ,P _(k,k−1) ^(xx|q) ^(k) )B(q _(k);μ_(k,k−1) ^(q) ^(k) ,1)μ_(k,k−1) ^(x|q) ^(k) =∫ƒ_(k)(x _(k−1))N(x _(k);μ_(k−1,k−1) ^(x|q) ^(k),P _(k−1,k−1) ^(xx|q) ^(k) )dx _(k−1)μ_(k,k−1) ^(q) ^(k) =M _(k)μ_(k−1,k−1) ^(q) ^(k−1)P _(k,k−1) ^(xx|q) ^(k) =Q _(k)+∫ƒ_(k)(x _(k−1))ƒ_(k) ^(T)(x _(k−1))N(x_(k);μ_(k−1,k−1) ^(x|q) ^(k) ,P _(k−1,k−1) ^(xx|q) ^(k) )dx_(k−1)−[μ_(k,k−1) ^(x|q) ^(k) ]^(T)μ_(k,k−1) ^(x|q) ^(k) .

To calculate these moments requires the solution of an M-dimensionalquadrature problem. However this formulation is similar to the Kalmanfilter, and depending on the nature of the target dynamics model manytechniques are available that provide accurate real-time solutions.

In the second step of the computational strategy, a target stateposterior conditioned on type,p(x _(k) ,q _(k) |T,z _(1,2, . . . , k)), wherep(x _(k) ,q _(k) |T,z _(1,2, . . . , k))=p(x _(k) |q _(k) ,T,z_(1,2, . . . , k))p(q _(k) |z _(1,2, . . . , k)),is determined. The second step is shown schematically in the stochasticsystem model diagram shown in FIG. 3 b.

The measurements are considered as a deterministic function of targetstate and type and corrupted by independent noise,p(z _(k) |x _(k) ,T)=N(z _(k) ;h _(k)(x _(k) ,T),R _(k))x_(k) & z_(k) are considered to be jointly Gaussian and approximatedwith the predictive density,

${p\left( {x_{k},\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)} \approx {N\left( {{{\begin{bmatrix}x_{k} \\z_{k}\end{bmatrix};}\begin{bmatrix}\mu_{k,{k - 1}}^{x|T} \\\mu_{k,{k - 1}}^{z|T}\end{bmatrix}},\begin{bmatrix}P_{k,{k - 1}}^{{xx}|T} & P_{k,{k - 1}}^{{xz}|T} \\P_{k,{k - 1}}^{{xz}|T} & P_{k,{k - 1}}^{{zz}|T}\end{bmatrix}} \right)}$

Where,μ_(k,k−1) ^(z|T) =∫h _(k)(x _(k) ,T)N(x _(k);μ_(k,k−1) ^(x|T) ,P_(k,k−1) ^(xx|T))dx _(k)P _(k,k−1) ^(zz|T) =R _(k) +∫h _(k)(x _(k) ,T)h _(k) ^(T)(x _(k) ,T)N(x_(k);μ_(k,k−1) ^(x|T) ,P _(k,k−1) ^(xx|T))dx _(k−1)−[μ_(k,k−1)^(z|T)]^(T)μ_(k,k−1) ^(z|T)P _(k,k−1) ^(xz|T) =∫x _(k) h _(k) ^(T)(x _(k) ,T)N(x _(k);μ_(k,k−1)^(x|T) ,P _(k,k−1) ^(xx|T))dx _(k−1)−μ_(k,k−1) ^(x|T)[μ_(k,k−1)^(z|T)]^(T)

Then the approximate conditional target state posterior density is,p(x _(k) |T,z _(1,2, . . . , k))≈N(x _(k);μ_(k,k−1) ^(x|T) +P _(k,k−1)^(xx|T)[P _(k,k−1) ^(zz|T)]⁻¹(z _(k)−μ_(k,k−1) ^(z|T)),P _(k,k−1)^(xx|T) −P _(k,k−1) ^(xz|T)[P _(k,k−1) ^(zz|T)]⁻¹ P _(k,k−1) ^(zx|T))

Similarly,

${p\left( q_{k} \middle| z_{1,2,\ldots,k} \right)} = {\frac{1}{\alpha}{p\left( {\left. z_{k} \middle| q_{k} \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( q_{k} \middle| z_{1,2,\ldots,{k - 1}} \right)}}$${p\left( q_{k} \middle| z_{1,2,\ldots,{k - 1}} \right)} = {\frac{1}{\alpha}{p\left( {\left. z_{k} \middle| q_{k} \right.,z_{1,2,\ldots,{k - 1}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}}$

Expanding the normalization,

${p\left( q_{k} \middle| z_{1,2,\ldots,k} \right)} = {\frac{{p\left( {\left. z_{k} \middle| q_{k} \right.,z_{1,2,\ldots,{k - 1}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}}{\sum\limits_{{{\overset{\sim}{q}}_{k} = 0},1}{{p\left( {\left. z_{k} \middle| {\overset{\sim}{q}}_{k} \right.,z_{1,2,\ldots,{k - 1}}} \right)}{B\left( {{{\overset{\sim}{q}}_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}}}.}$

As shown previously above, the result includes an M dimensional integralof the formI=∫ƒ(x)N(x;μ ^(x) ,P ^(xx))dx

At this stage of the calculation, measurement data is being employed torefine hidden target state estimates, for which computationallyefficient numerical methods are available (these may be solved preciselyup to the nth order with an n knot Hermite polynomial approximation).

In the third step of the computational strategy, a final estimate of thetarget type density is generated. The third step is shown schematicallyin the stochastic system model diagram shown in FIG. 3 c.

Inductively starting from

p(T❘z_(1, 2, …  , k − 1)), then${p\left( {T❘z_{1,2,\ldots\mspace{11mu},k}} \right)} = \frac{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},{\hat{T}❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}}$$\begin{matrix}{{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)} = {{p\left( {{z_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}\sum\limits_{{q_{k} = 0},1}}} \\{\int{{p\left( {z_{k},x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{p\left( {{z_{k}❘T},x_{k},q_{k}} \right)}}}}} \\{{p\left( {x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{p\left( {{z_{k}❘T},x_{k},q_{k}} \right)}}}}} \\{{N\left( {{x_{k};\mu_{k,{k - 1}}^{x❘q_{k}}},P_{k,{k - 1}}^{{xx}❘q_{k}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}d\;{x_{k}.}}\end{matrix}$

Traditional methods do not yield sufficiently accurate solutions to thisintegral because non-linear functions with ‘narrow’ modes breakband-limited quadrature schemes such as the unscented transform andsequential Monte Carlo (e.g. particle filters). An adaptive quadratureand subspace partitioning scheme of these teachings is disclosed hereinbelow. The adaptive quadrature and subspace partitioning scheme of theseteachings includes subspace partitioning, whitening and mixedquadrature, the details of which are disclosed herein below.

Subspace Partitioning

An accurate real-time calculation of an M-dimensional integral of theform,I=∫p(z|x,T,q)N(x;μ ^(x) ,P ^(xx))dxis desired.

Variable indexing is dropped for notational convenience in thediscussion of quadrature. Denote each variate of x: x=[x₁, x₂, . . .x_(N)]^(T) and each variate of z: z=[z₁, z₂, . . . z_(F)]^(T).

Measurement Fisher information is defined as,

${F_{i,j}\left( {x,T,q} \right)} = {\int{\left( {\frac{\partial}{\partial x_{i}}\log\;{p\left( {{z❘x},T,q} \right)}\frac{\partial}{\partial x_{j}}\log\;{p\left( {{z❘x},T,q} \right)}} \right){p\left( {{z❘x},T,q} \right)}d\; z}}$

By construction the measurement model is a deterministic function hcorrupted by white additive noise with covariance R, so the Fisherinformation of the i-th and j-th variates of x becomes,

${F_{i,j}\left( {x,T,q} \right)} = {\prod\limits_{k}{\int{\frac{\partial}{{\partial x_{i}}x_{j}}\log\;{p\left( {{z_{k}❘x},T,q} \right)}{p\left( {{z_{k}❘x},T,q} \right)}d\; z}}}$

A further simplification can be made during implementation since byconstruction each variate of z is independent given x and T and R isdiagonal.

Analytically or numerically compute the Fisher information matrix F.Depending on the specific form of the problem domain one either letsx=μ^(x) or approximately marginalizes out x in F, making thiscalculation computationally efficient.

Now, by a numerical method such as singular value decomposition, ∃U, V,S∈

^(NXN) s.t. U & V are unitary, S is diagonal, S_(ii)≥S_(jj) for i>j, andF=USV^(T)

$s = {{\begin{matrix}\max \\i\end{matrix}S_{ii}} \geq {ɛ\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}$

Denote the i-th column of U by U_(i) and define Û=[U₁, . . . , U_(s)]

Define a new s-dimensional variate w=Ûx, reducing I to an s-dimensionalintegral

$\begin{matrix}{I = {\int{{p\left( {{z❘{\hat{U}w}},T,q} \right)}{N\left( {{w;{{\hat{U}}^{T}\mu^{x}}},{{\hat{U}}^{T}P^{xx}\hat{U}}} \right)}d\; w}}} \\{= {\int{{p\left( {{z❘{\hat{U}w}},T,q} \right)}{N\left( {{w;\mu^{w}},P^{ww}} \right)}d\; w}}}\end{matrix}$Whitening

There is a Ũ in

^(SXS) such that Ũ is lower triangular and Ũ^(T)Ũ=P^(ww)

Define a new variate v

$v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}$

Then we have

$I = {\frac{\sqrt{2}}{\left( {2\pi} \right)^{\frac{s}{2}}}{\int{{p\left( {{z❘{\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)}},T,q} \right)}e^{{- v^{T}}v}d\; v}}}$Mixed Quadrature

The transformed Fisher information is {circumflex over(F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹

Define q′₁={i:{circumflex over (F)}_(ii)≤α}, q′₂={i:{circumflex over(F)}_(ii)>α} where α is a constant

Employ Hermite approximation for q′₁ variates & brute force numericaltechniques for q′₂

∀i∈q′₁ select Hermite order as ┌β{circumflex over (F)}_(ii)┐ where β isa constant yielding weights w_(i) and knots k_(i)

∀i∈q′₂ select quadrature order as

$\left\lceil \frac{{\hat{F}}_{ii}}{ϰ} \right\rceil$where χ is a constant yielding weights w_(i) and knots k_(i)

Let

denote the column-wise Khatri-Rao product and Î_(i) a sxi column vectorof ones

  w = w₁ ⊗ w₂ ⊗ … ⊗ w_(s)  k = [Î₀k₁Î_(s − 1)] ⊗ [Î₁k₂Î_(s − 2)] ⊗ … ⊗ [Î_(s − 1)k_(s)Î₀]$I = {{\frac{\sqrt{2}}{\left( {2\pi} \right)^{\frac{s}{2}}}{\int{{p\left( {{z❘{\hat{U}\left( {{\sqrt{2}\hat{U}v} + \mu^{w}} \right)}},T,q} \right)}e^{{- v^{T}}v}d\; v}}} \approx {\sum\limits_{{i = 1},{\ldots\mspace{14mu}{w}}}{{p\left( {{z❘{\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)}},T,q} \right)}w_{i}}}}$

To illustrate the performance gains of the method of these teachings ascompared to conventional target identification methods, areduced-dimensional problem whose states are easy to convey graphicallyis disclosed herein below. In this example, presume the observer andtarget are embedded on the real line with the observer stationary at theorigin—this represents simplification of observer-to-target line ofsight. The target state is fully described by its range and‘orientation’ with respect to the line of sight. The target velocity isconstant, however its speed along the line of sight varies with thecosine of its orientation. Both target states are observable, butseverely corrupted by noise. Finally, the wingspan of the targetprojected onto plane normal to the line-of-sight (which is its wingspanscaled by the cosine of its orientation) is observable as an ID feature.

An embodiment of the method for adaptive quadrature of these teachingsis summarized in FIG. 4. Referring to FIG. 4, in the embodiment 302shown there in, adaptive subspace partitioning 402 is first performed,followed by whitening 404, and then followed by adaptive mixedquadrature rule selection 406. Finally, the mixed quadrature result 410is obtained.

The entire computational strategy disclosed hereinabove is summarized inFIG. 5. Referring to FIG. 5, the target type dependent dynamics model306 is used to provide evolution from the hidden state at one look ortime, p(x_(k−1)|T, z_(1, 2, . . . , k−1)) to the hidden states at thesubsequent look or time, p(x_(k)|T, z_(1, 2, . . . , k)). The time delay304 can be symbolic, showing the relationship between the two looks ortimes. The measurements, z_(k), the hidden states and the target type,x_(k),T, together with the output from the feature model 310 arecombined in the likelihood function 312 in order to providep(z_(k)|x_(k),T). The two conditional probabilities, p(z_(k)|x_(k),T)and p(x_(k)|T, z_(1, 2, . . . , k)), as well asp(T|z_(1, 2, . . . , k−1)) are provided to the adaptive quadraturemethod 302 in order to obtain p(T|z_(1, 2, . . . , k)), the conditionalprobability of the target type given the measurements. Finally,p(T|z_(1, 2, . . . , k)) is provided to the decision rule 314 in orderto obtain an estimate of the target type.

In order to illustrate these teachings, results for simple example ofprovided herein below. This example serves to motivate elements of themethod. The example relates to discriminating between two different typeof flying targets, using four notational CID range features (1σ=1.0 m).The measurement failure is modeled as a Markov process (P(qk=qk−1)=0.8).The targets are considered to be 50 km away flying straight and level at200 m/s. The fuselage and wings of two types of targets are modeled with3-D rectangles. The results are shown in FIG. 6 illustrate theimportance of including measurement degradation (q_(k) estimated) versusnot including measurement degradation (q_(k) not estimated).

The following is a disclosure by way of example of a device configuredto execute functions (hereinafter referred to as computing device) whichmay be used with the presently disclosed subject matter. The descriptionof the various components of a computing device is not intended torepresent any particular architecture or manner of interconnecting thecomponents. Other systems that have fewer or more components may also beused with the disclosed subject matter. A communication device mayconstitute a form of a computing device and may at least include acomputing device. The computing device may include an interconnect(e.g., bus and system core logic), which can interconnect suchcomponents of a computing device to a data processing device, such as aprocessor(s) or microprocessor(s), or other form of partly or completelyprogrammable or pre-programmed device, e.g., hard wired and or ASICcustomized logic circuitry, such as a controller or microcontroller, adigital signal processor, or any other form of device that can fetchinstructions, operate on pre-loaded/pre-programmed instructions, and/orfollowed instructions found in hard-wired or customized circuitry tocarry out logic operations that, together, perform steps of and wholeprocesses and functionalities as described in the present disclosure.

In this description, various functions, functionalities and/oroperations may be described as being performed by or caused by softwareprogram code to simplify description. However, those skilled in the artwill recognize what is meant by such expressions is that the functionsresult from execution of the program code/instructions by a computingdevice as described above, e.g., including a processor, such as amicroprocessor, microcontroller, logic circuit or the like.Alternatively, or in combination, the functions and operations can beimplemented using special purpose circuitry, with or without softwareinstructions, such as using ASIC or FPGA, which may be programmable,partly programmable or hard wired. The ASIC logic may be such as gatearrays or standard cells, or the like, implementing customized logic bymetallization(s) interconnects of the base gate array ASIC architectureor selecting and providing metallization(s) interconnects betweenstandard cell functional blocks included in a manufacturer's library offunctional blocks, etc. Embodiments can thus be implemented usinghardwired circuitry without program software code/instructions, or incombination with circuitry using programmed software code/instructions.

Thus, the techniques are limited neither to any specific combination ofhardware circuitry and software, nor to any particular tangible sourcefor the instructions executed by the data processor(s) within thecomputing device. While some embodiments can be implemented in fullyfunctioning computers and computer systems, various embodiments arecapable of being distributed as a computing device including, e.g., avariety of forms and capable of being applied regardless of theparticular type of machine or tangible computer-readable media used toactually effect the performance of the functions and operations and/orthe distribution of the performance of the functions, functionalitiesand/or operations.

The interconnect may connect the data processing device to define logiccircuitry including memory. The interconnect may be internal to the dataprocessing device, such as coupling a microprocessor to on-board cachememory or external (to the microprocessor) memory such as main memory,or a disk drive or external to the computing device, such as a remotememory, a disc farm or other mass storage device, etc. Commerciallyavailable microprocessors, one or more of which could be a computingdevice or part of a computing device, include a PA-RISC seriesmicroprocessor from Hewlett-Packard Company, an 80×86 or Pentium seriesmicroprocessor from Intel Corporation, a PowerPC microprocessor fromIBM, a Sparc microprocessor from Sun Microsystems, Inc, or a 68xxxseries microprocessor from Motorola Corporation as examples.

The interconnect in addition to interconnecting devices such asmicroprocessor(s) and memory may also interconnect such elements to adisplay controller and display device, and/or to other peripheraldevices such as input/output (I/O) devices, e.g., through aninput/output controller(s). Typical I/O devices can include a mouse, akeyboard(s), a modem(s), a network interface(s), printers, scanners,video cameras and other devices which are well known in the art. Theinterconnect may include one or more buses connected to one anotherthrough various bridges, controllers and/or adapters. In one embodimentthe I/O controller includes a Universal Serial Bus (USB) adapter forcontrolling USB peripherals, and/or an IEEE-1394 bus adapter forcontrolling IEEE-1394 peripherals.

The memory may include any tangible computer-readable media, which mayinclude but are not limited to recordable and non-recordable type mediasuch as volatile and non-volatile memory devices, such as volatileRandom Access Memory (RAM), typically implemented as dynamic RAM (DRAM)which requires power continually in order to refresh or maintain thedata in the memory, and non-volatile Read Only Memory (ROM), and othertypes of non-volatile memory, such as a hard drive, flash memory,detachable memory stick, etc. Non-volatile memory typically may includea magnetic hard drive, a magnetic optical drive, or an optical drive(e.g., a DVD RAM, a CD ROM, a DVD or a CD), or other type of memorysystem which maintains data even after power is removed from the system.

A server could be made up of one or more computing devices. Servers canbe utilized, e.g., in a network to host a network database, computenecessary variables and information from information in the database(s),store and recover information from the database(s), track informationand variables, provide interfaces for uploading and downloadinginformation and variables, and/or sort or otherwise manipulateinformation and data from the database(s). In one embodiment a servercan be used in conjunction with other computing devices positionedlocally or remotely to perform certain calculations and other functionsas may be mentioned in the present application. At least some aspects ofthe disclosed subject matter can be embodied, at least in part,utilizing programmed software code/instructions. That is, the functions,functionalities and/or operations techniques may be carried out in acomputing device or other data processing system in response to itsprocessor, such as a microprocessor, executing sequences of instructionscontained in a memory, such as ROM, volatile RAM, non-volatile memory,cache or a remote storage device. In general, the routines executed toimplement the embodiments of the disclosed subject matter may beimplemented as part of an operating system or a specific application,component, program, object, module or sequence of instructions usuallyreferred to as “computer programs,” or “software.” The computer programstypically comprise instructions stored at various times in varioustangible memory and storage devices in a computing device, such as incache memory, main memory, internal or external disk drives, and otherremote storage devices, such as a disc farm, and when read and executedby a processor(s) in the computing device, cause the computing device toperform a method(s), e.g., process and operation steps to execute anelement(s) as part of some aspect(s) of the method(s) of the disclosedsubject matter.

A tangible machine readable medium can be used to store software anddata that, when executed by a computing device, causes the computingdevice to perform a method(s) as may be recited in one or moreaccompanying claims defining the disclosed subject matter. The tangiblemachine readable medium may include storage of the executable softwareprogram code/instructions and data in various tangible locations,including for example ROM, volatile RAM, non-volatile memory and/orcache. Portions of this program software code/instructions and/or datamay be stored in any one of these storage devices. Further, the programsoftware code/instructions can be obtained from remote storage,including, e.g., through centralized servers or peer to peer networksand the like. Different portions of the software programcode/instructions and data can be obtained at different times and indifferent communication sessions or in a same communication session.

The software program code/instructions and data can be obtained in theirentirety prior to the execution of a respective software application bythe computing device. Alternatively, portions of the software programcode/instructions and data can be obtained dynamically, e.g., just intime, when needed for execution. Alternatively, some combination ofthese ways of obtaining the software program code/instructions and datamay occur, e.g., for different applications, components, programs,objects, modules, routines or other sequences of instructions ororganization of sequences of instructions, by way of example. Thus, itis not required that the data and instructions be on a single machinereadable medium in entirety at any particular instance of time.

In general, a tangible machine readable medium includes any tangiblemechanism that provides (i.e., stores) information in a form accessibleby a machine (i.e., a computing device, which may be included, e.g., ina communication device, a network device, a personal digital assistant,a mobile communication device, whether or not able to download and runapplications from the communication network, such as the Internet, e.g.,an iPhone, Blackberry, Droid or the like, a manufacturing tool, or anyother device including a computing device, comprising one or more dataprocessors, etc.

For the purposes of describing and defining the present teachings, it isnoted that the term “substantially” is utilized herein to represent theinherent degree of uncertainty that may be attributed to anyquantitative comparison, value, measurement, or other representation.The term “substantially” is also utilized herein to represent the degreeby which a quantitative representation may vary from a stated referencewithout resulting in a change in the basic function of the subjectmatter at issue.

Although the invention has been described with respect to variousembodiments, it should be realized these teachings are also capable of awide variety of further and other embodiments within the spirit andscope of the appended claims.

What is claimed is:
 1. A target identification system comprising: amemory for storing data comprising: a data structure stored in thememory, the data structure including: a number of measurements of atarget, each measurement from the number of measurements being observedat a predetermined time (z_(k)); a number of target types (T); each oneof the number of measurements, each one the number of target type andeach one of one or more hidden states, each hidden state (x_(k)) beingcharacterized at the predetermined time, being correlated to oneanother; one or more processors operatively connected to the memory forstoring data; the one or more processors being configured to: provide afirst conditional probability distribution, a conditional probability ofa target type given a number of measurements, defined inductively by${{p\left( {T❘z_{1,2,\ldots\mspace{11mu},k}} \right)} = \frac{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},{\hat{T}❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}}},{and}$$\begin{matrix}{{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)} = {{p\left( {{z_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}\sum\limits_{{q_{k} = 0},1}}} \\{\int{{p\left( {z_{k},x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{p\left( {{z_{k}❘T},x_{k},q_{k}} \right)}}}}} \\{{p\left( {x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}}\end{matrix}$ where p(z_(k,i)|T,x_(k)) is a conditional probability ofan ith measurement at a kth instance giving a target type T and hiddenstates at the kth instance, and p(T|z_(1, 2, . . . k−1)) is aconditional probability distribution function of a target type, T, giventhe plurality of measurements at the k−1^(th) time, z_(1, . . . k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0; p(z_(k), x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of a measurement at the kth instance, hidden states at the kthinstance, measurement quality at the kth instance given a target typeand the plurality of measurements at the k−1^(th) time,z_(1, . . . k−1); p(z_(k)|T, x_(k), q_(k)) is a conditional probabilitydistribution function of the measurement at the kth instance given atarget type, the hidden states at the kth instance and the measurementquality at the kth instance; and p(x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of the measurement at the kth instance and the measurementquality at the kth instance given a target type and the plurality ofmeasurements at the k−1^(th) time, z_(1, . . . k−1); and obtain anestimate of the target type from the first conditional probability. 2.The target identification system of claim 1 further comprising anon-transitory computer usable media having computer readable codeembodied therein, wherein the computer readable code, when executed bythe one or more processors, causes the one or more processors to:provide a first conditional probability distribution, a conditionalprobability of a target type given a number of measurements, definedinductively by${{p\left( {T❘z_{1,2,\ldots\mspace{11mu},k}} \right)} = \frac{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},{\hat{T}❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)}}},{and}$$\begin{matrix}{{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)} = {{p\left( {{z_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}\sum\limits_{{q_{k} = 0},1}}} \\{\int{{p\left( {z_{k},x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{p\left( {{z_{k}❘T},x_{k},q_{k}} \right)}}}}} \\{{p\left( {x_{k},{q_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}d\; x_{k}}\end{matrix}$ where p(z_(k,i)|T,x_(k)) is a conditional probability ofan ith measurement at a kth instance giving a target type T and hiddenstates at the kth instance, and p(T|z_(1, 2, k−1)) is a conditionalprobability distribution function of a target type, T, given theplurality of measurements at the k−1^(th) time, z_(1, k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0; p(z_(k), x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of a measurement at the kth instance, hidden states at the kthinstance, measurement quality at the kth instance given a target typeand the plurality of measurements at the k−1^(th) time,z_(1, . . . k−1); p(z_(k)|T, x_(k), q_(k)) is a conditional probabilitydistribution function of the measurement at the kth instance given atarget type, the hidden states at the kth instance and the measurementquality at the kth instance; and p(x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of the measurement at the kth instance and the measurementquality at the kth instance given a target type and the plurality ofmeasurements at the k−1^(th) time, z_(1, . . . k−1); and obtain anestimate of the target type from the first conditional probability;thereby configuring the one or more processors.
 3. The targetidentification system of claim 1 wherein transition between one hiddenstate at one instance and the one hidden state at another instance isgiven by a predetermined dynamic model (referred to as ƒ_(k)(x_(k−1),T)); whereinp(x _(k) |x _(k−1) ,T)=N(x _(k);ƒ_(k)(x _(k−1) ,T),Q _(k)) where N(⋅; u,P) denotes a multivariate Gaussian with mean u and covariance P; whereinmeasurement quality is considered to be described by a Bernoullidistribution, $\begin{matrix}{{p\left( {{q_{k}❘q_{k - 1}},T} \right)} = {{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)} = \left\{ {\begin{matrix}{M_{k}q_{k - 1}} & {q_{k} = q_{k - 1}} \\{\left( {1 - M_{k}} \right)q_{k - 1}} & {q_{k} \neq q_{k - 1}}\end{matrix};} \right.}} & \;\end{matrix}$ and wherein an expectation of p(x_(k)|T,z_(1, 2, . . . k−1)) is given byμ_(k,k−1) ^(x|q) ^(k) =∫ƒ_(k)(x _(k−1))N(x _(k);μ_(k−1,k−1) ^(x|q) ^(k),P _(k−1,k−1) ^(xx|q) ^(k) )dx _(k−1) and a covariance matrix is giveninductively byP _(k,k−1) ^(xx|q) ^(k) =Q _(k)+∫ƒ_(k)(x _(k−1))ƒ_(k) ^(T)(x _(k−1))N(x_(k);μ_(k−1,k−1) ^(x|q) ^(k) ,P _(k−1,k−1) ^(xx|q) ^(k) )dx_(k−1)−[μ_(k,k−1) ^(x|q) ^(k) ]^(T)μ_(k,k−1) ^(x|q) ^(k) .
 4. The targetidentification system of claim 1 wherein p(x_(k)|x_(k−1),T) is amultivariate Gaussian distribution and wherein p(x_(k)|T, z_(k)) isanother multivariate Gaussian distribution, where p(x_(k)|x_(k−1),T) isa conditional probability distribution of hidden states at instance kgiven hidden states at instance k−1 and target type T; and whereinp(q_(k)|q_(k−1), T) is a Bernoulli distribution, B(q_(k); q_(k−1),M_(k)), and${B\left( {{q_{k};q_{k - 1}},M_{k}} \right)} = \left\{ {\begin{matrix}{M_{k}q_{k - 1}} & {q_{k} = q_{k - 1}} \\{\left( {1 - M_{k}} \right)q_{k - 1}} & {q_{k} \neq q_{k - 1}}\end{matrix}.} \right.$
 5. The target identification system of claim 4wherein p(T|z_(1, 2, . . . , k)) is approximated by adaptive quadrature;and wherein$\mspace{20mu}{{{p\left( {T❘z_{1,2,{\ldots\mspace{14mu} k}}} \right)} = {{p\left( {T,{z_{k}❘z_{1,2,{{\ldots\mspace{14mu} k} - 1}}}} \right)}/\underset{T}{\Sigma}}},{p\left( {T^{\prime},{z_{k}❘z_{1,2,{{\ldots\mspace{14mu} k} - 1}}}} \right)},\mspace{20mu}{w\;{here}}}$$\begin{matrix}{\mspace{79mu}{{p\left( {z_{k},{T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}}} \right)} = {{p\left( {{z_{k}❘T},z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}}}} \\{= {{p\left( {T❘z_{1,2,\ldots\mspace{11mu},{k - 1}}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{p\left( {{z_{k}❘T},x_{k},q_{k}} \right)}}}}} \\{{N\left( {{x_{k};\mu_{k,{k - 1}}^{x❘q_{k}}},P_{k,{k - 1}}^{{xx}❘q_{k}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}d\; x_{k}}\end{matrix}$   I = ∫p(z❘x, T, q)N(x; μ^(x), P^(xx))d x$I = {{\frac{\sqrt{2}}{\left( {2\pi} \right)^{\frac{s}{2}}}{\int{{p\left( {{z❘{\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)}},T,q} \right)}e^{{- v^{T}}v}d\; v}}} \approx {\sum\limits_{{i = 1},{\ldots\mspace{14mu}{w}}}{{p\left( {{z❘{\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)}},T,q} \right)}w_{i}}}}$where p(T, z_(k)|z_(1, 2, . . . k−1)) is a second conditionalprobability, p(z_(k)|T, x_(k)) is a third conditional probability, aFisher information is transformed by F=USV^(T), U & V are unitary, S isdiagonal, S_(ii)≥S_(jj) for i>j, ${s = {{\begin{matrix}\max \\i\end{matrix}\mspace{14mu} S_{ii}} \geq {ɛ\mspace{20mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)], w=Ûx,μ^(w) is an expectation of w at step i, P^(ww) is an expectation ofww^(T), Ũ is lower triangular and Ũ^(T)Ũ=P^(ww), a new variable v isdefined by${v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}},$a transformed Fisher information is {circumflex over(F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹, q₁={i:{circumflex over (F)}_(ii)≤α},q₂={i:{circumflex over (F)}_(ii)>α} where α is a constant, for each i inq₁ select Hermite order as |β{circumflex over (F)}_(ii)| where β is aconstant yielding weights w_(i) and knots k_(i), for each i in q₂ selectquadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\mathcal{X}} \right\rceil$ where χ isa constant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀].6. The target identification system of claim 2 whereinp(T|z_(1, 2, . . . , k)) is approximated by adaptive quadrature; andwherein$\mspace{20mu}{{{p\left( T \middle| z_{1,2,{\ldots\mspace{14mu} k}} \right)} = {{p\left( {T,\left. z_{k} \middle| z_{1,2,{{\ldots\mspace{14mu} k} - 1}} \right.} \right)}/\sum\limits_{T}^{\;}}},{p\left( {T^{\prime},\left. z_{k} \middle| z_{1,2,{{\ldots\mspace{14mu} k} - 1}} \right.} \right)},\mspace{20mu}{where}}$${p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)} = {{{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}} = {{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}^{\;}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{N\left( {{x_{k};\mu_{k,{k - 1}}^{x|q_{k}}},p_{k,{k - 1}}^{{xx}|q_{k}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}d\; x_{k}}}}}}$  I = ∫p(z|x, T, q)N(x; μ^(x), P^(xx))d x$I = {{\frac{\sqrt{2}}{\left( {2\pi} \right)^{s/2}}{\int{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)} \right.,T,q} \right)}e^{{- v^{T}}v}d\; v}}} \approx {\sum\limits_{{i = 1},{\ldots\mspace{14mu}{w}}}^{\;}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T,q} \right)}w_{i}}}}$where p(T|z_(1, 2, . . . k−1)) is a second conditional probability,p(z_(k)|T, x_(k)) is a third conditional probability, a Fisherinformation is transformed by F=USV^(T), U & V are unitary, S isdiagonal, S_(ii)≥S_(jj) for i>j, ${s = {{\begin{matrix}\max \\i\end{matrix}S_{ii}} \geq {ɛ\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)], w=Ûx,μ^(w) is an expectation of w at step i, P^(ww) is an expectation ofww^(T), Ũ is lower triangular and Ũ^(T)Ũ=P^(ww), a new variable v isdefined by${v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}},$a transformed Fisher information is {circumflex over(F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹, q₁={i:{circumflex over (F)}_(ii)≤α},q₂={i:{circumflex over (F)}_(ii)>α} where α is a constant, for each i inq₁ select Hermite order as ┌β{circumflex over (F)}_(ii)┐ where β is aconstant yielding weights w_(i) and knots k_(i), for each i in q₂ selectquadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\mathcal{X}} \right\rceil$ where χ isa constant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀].7. The target identification system of claim 1 wherein the estimate isobtained using a Bayes classifier.
 8. The target identification systemof claim 1, wherein the measurements z_(i) comprise one or more of arange to the target, a rate of change of a range to the target, anacceleration along a line of sight.
 9. The target identification systemof claim 1, wherein the hidden states x_(i) comprise one or more oftarget geometry, target attitude, target velocity.
 10. The targetidentification system of claim 1, wherein the plurality of target typescomprise one or more of types of vehicles.
 11. A method for targetidentification comprising: receiving, at one or more processors, anumber of measurements of a target, each measurement from the number ofmeasurements being observed at a predetermined time (z_(k)), a number oftarget types (T); each one of the number of measurements, each one thenumber of target type and each one of one or more hidden states, eachhidden state (x_(k)) being characterized at the predetermined time,being correlated to one another; providing, using the one or moreprocessors, a first conditional probability distribution, a conditionalprobability of a target type given a number of measurements, definedinductively by$\mspace{20mu}{{{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}^{\;}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)}}},\mspace{20mu}{and}}$${p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)} = {{{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}} = {{{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}^{\;}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}d\; x_{k}}}}} = {{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}^{\;}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}d\; x_{k}}}}}}}$where p(z_(k,i)|T,x_(k)) is a conditional probability of an ithmeasurement at a kth instance giving a target type T and hidden statesat the kth instance, and p(T|z_(1, 2, . . . k−1)) is a conditionalprobability distribution function of a target type, T, given theplurality of measurements at the k−1^(th) time, z_(1, . . . k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0; p(z_(k), x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of a measurement at the kth instance, hidden states at the kthinstance, measurement quality at the kth instance given a target typeand the plurality of measurements at the k−1^(th) time,z_(1, . . . k−1); p(z_(k)|T, x_(k), q_(k)) is a conditional probabilitydistribution function of the measurement at the kth instance given atarget type, the hidden states at the kth instance and the measurementquality at the kth instance; and p(x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of the measurement at the kth instance and the measurementquality at the kth instance given a target type and the plurality ofmeasurements at the k−1^(th) time, z_(1, . . . k−1); and obtaining anestimate of the target type from the first conditional probability;wherein the providing a first conditional probability distribution andthe obtaining an estimate of the target type are performed by the one ormore processors executing computer readable code embodied innon-transitory computer usable media.
 12. The method of claim 11 whereinp(x_(k)|x_(k−1),T) is a multivariate Gaussian distribution given byp(x _(k) |x _(k−1) ,T)=N(x _(k);ƒ_(k)(x _(k−1) ,T),Q _(k)) where N(⋅; u,P) denotes a multivariate Gaussian with mean u and covariance P; andwherein p(x_(k)|T, z_(k)) is another multivariate Gaussian distribution,where p(x_(k)|x_(k−1),T) is a conditional probability distribution ofhidden states at instance k given hidden states at instance k−1 andtarget type T.
 13. The method of claim 12 wherein transition between onehidden state at one instance and the one hidden state at anotherinstance is given by a predetermined dynamic model (referred to asƒ_(k)(x_(k−1), T)); and wherein measurement quality is considered to bedescribed by a Bernoulli distribution,${p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)} = {{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)} = \left\{ {\begin{matrix}{M_{k}q_{k - 1}} & {q_{k} = q_{k - 1}} \\{\left( {1 - M_{k}} \right)q_{k - 1}} & {q_{k} \neq q_{k - 1}}\end{matrix};} \right.}$ and wherein an expectation of p(x_(k)|T,z_(1, 2, . . . k−1)) is given byμ_(k,k−1) ^(x|q) ^(k) =∫ƒ_(k)(x _(k−1))N(x _(k);μ_(k−1,k−1) ^(x|q) ^(k),P _(k−1,k−1) ^(xx|q) ^(k) )dx _(k−1) and a covariance matrix is giveninductively byP _(k,k−1) ^(xx|q) ^(k) =Q _(k)+∫ƒ_(k)(x _(k−1))ƒ_(k) ^(T)(x _(k−1))N(x_(k);μ_(k−1,k−1) ^(x|q) ^(k) ,P _(k−1,k−1) ^(xx|q) ^(k) )dx_(k−1)−[μ_(k,k−1) ^(x|q) ^(k) ]^(T)μ_(k,k−1) ^(x|q) ^(k) .
 14. Themethod of claim 13 wherein measurements are a predetermined function oftarget state and type (referred to as h_(k)(x_(k), T)) corrupted byindependent noise; and wherein p(x_(k)|T, z_(1, 2, . . . k−1)) is amultivariate Gaussian distribution.
 15. The method of claim 13 whereinp(T|z_(1, 2, . . . , k)) is approximated by adaptive quadrature; andwherein$\mspace{20mu}{{{p\left( T \middle| z_{1,2,{\ldots\mspace{14mu} k}} \right)} = {{p\left( {T,\left. z_{k} \middle| z_{1,2,{{\ldots\mspace{14mu} k} - 1}} \right.} \right)}/\sum\limits_{T}^{\;}}},{p\left( {T^{\prime},\left. z_{k} \middle| z_{1,2,{{\ldots\mspace{14mu} k} - 1}} \right.} \right)},\mspace{20mu}{where}}$${p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)} = {{{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}} = {{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}^{\;}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{N\left( {{x_{k};\mu_{k,{k - 1}}^{x|q_{k}}},p_{k,{k - 1}}^{{xx}|q_{k}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}d\; x_{k}}}}}}$  I = ∫p(z|x, T, q)N(x; μ^(x), P^(xx))d x$I = {{\frac{\sqrt{2}}{\left( {2\pi} \right)^{s/2}}{\int{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)} \right.,T,q} \right)}e^{{- v^{T}}v}d\; v}}} \approx {\sum\limits_{{i = 1},{\ldots\mspace{14mu}{w}}}^{\;}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T,q} \right)}w_{i}}}}$where p(T, z_(k)|z_(1, 2, . . . k−1)) is a second conditionalprobability, p(z_(k)|T, x_(k)) is a third conditional probability, aFisher information is transformed by F=USV^(T), U & V are unitary, S isdiagonal, S_(ii)≥S_(jj) for i>j, ${s = {{\begin{matrix}\max \\i\end{matrix}S_{ii}} \geq {ɛ\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)], w=Ûx,μ^(w) is an expectation of w at step i, P^(ww) is an expectation ofww^(T), Ũ is lower triangular and Ũ^(T)Ũ=P^(ww), a new variable v isdefined by${v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}},$a transformed Fisher information is {circumflex over(F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹, q₁={i:{circumflex over (F)}_(ii)≤α},q₂={i:{circumflex over (F)}_(ii)>α} where α is a constant, ∀i∈q₁ selectHermite order as |β{circumflex over (F)}_(ii)| where β is a constantyielding weights w_(i) and knots k_(i), ∀i∈q₂ select quadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\mathcal{X}} \right\rceil$ where χ isa constant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀].16. The method of claim 11 wherein the estimate is obtained using aBayes classifier.
 17. The method of claim 15 wherein providing a firstconditional probability comprises: inductively obtaining hidden stateinformation at a kth stage from hidden state information at a preceding(k−1)th stage; a obtaining a fourth conditional probability of a hiddenstate given a target type and a number of measurements, p(x_(k),q_(k)|T, z_(1, 2, . . . , k−1)) and obtaining the first conditionalprobability, expressed inductively, as a function of the fourthconditional probability.
 18. The method of claim 15 wherein in defininginductively the first conditional probability as$\mspace{20mu}{{{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}^{\;}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)}}},{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right.} \right)} = {{p\left( T \middle| z_{1,2,\ldots\mspace{14mu},{k - 1}} \right)}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k}} \right)}{p\left( {\left. x_{k} \middle| T \right.,z_{1,2,\ldots\mspace{14mu},{k - 1}}} \right)}d\;{x_{k}.}}}}}}$19. The method of claim 18 wherein the third and fourth conditionalprobabilities are represented by multivariate Gaussian probabilityfunctions.
 20. The method of claim 19 wherein providing the firstconditional probability distribution further comprises: using subspacepartitioning to reduce dimensions of integration variable space ofintegralI=∫p(z|x,T,q)N(x;μ ^(x) ,P ^(xx))dx; transforming variables in theintegral in order to apply whitening; and applying differentapproximation methods to regions of more importance and regions of lessimportance, importance measured by Fisher information, in order toapproximate the integral by a sum,$\sum\limits_{{i = 1},{\ldots\mspace{14mu}{w}}}^{\;}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T,q} \right)}{w_{i}.}}$21. The method of claim 20 wherein using subspace partitioningcomprises: transforming the integration variable space by w=Ûx, where aFisher information is transformed by F=USV^(T), U & V are unitary, S isdiagonal, S_(ii)≥S_(jj) for i>j, ${s = {{\begin{matrix}\max \\i\end{matrix}S_{ii}} \geq {ɛ\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$and an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)].22. The method of claim 21 wherein transforming variables in theintegral in order to apply whitening comprises: transforming from w to anew variable v defined by$v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}$in order to apply whitening.
 23. The method of claim 22 wherein atransformed Fisher information is {circumflex over (F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹,q₁={i:{circumflex over (F)}_(ii)≤α}, q₂={i:{circumflex over (F)}_(ii)>α}where α is a constant, for each i in q₁ select Hermite order as┌β{circumflex over (F)}_(ii)┐ where β is a constant yielding weightsw_(i) and knots k_(i), for each i in q₂ select quadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\chi} \right\rceil$  where χ is aconstant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀]and wherein${\int{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)} \right.,T} \right)}e^{{- v^{T}}v}d\; v}} \approx {\sum\limits_{{i = 1},{\ldots|w|}}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T} \right)}{w_{i}.}}}$24. A non-transitory computer readable medium having computer executablecode embodied therein, said computer executable code, when executed inat least one processor, causing the at least one processor to: receive,at the at least one processor, a number of measurements of a target,each measurement from the number of measurements being observed at apredetermined time (z_(k)), a number of target types (T); each one ofthe number of measurements, each one the number of target type and eachone of one or more hidden states, each hidden state (x_(k)) beingcharacterized at the predetermined time, being correlated to oneanother; provide a first conditional probability distribution, aconditional probability of a target type given a number of measurements,defined inductively by provide a first conditional probabilitydistribution, a conditional probability of a target type given a numberof measurements, defined inductively by${{p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}},{and}$$\begin{matrix}{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {z_{k},x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{p\left( {x_{k},\left. q_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\; x_{k}}}}}}\end{matrix}$ where p(z_(k,i)|T,x_(k)) is a conditional probability ofan ith measurement at a kth instance giving a target type T and hiddenstates at the kth instance, and p(T|z_(1, 2, . . . k−1)) is aconditional probability distribution function of a target type, T, giventhe plurality of measurements at the k−1^(th) time, z_(1, . . . k−1);q_(l)={q_(l,1), q_(l,2), . . . , q_(l,F)}: measurement quality at lookl, E{q_(l,i)q_(l,j)}_(i≠j)=0; p(z_(k), x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of a measurement at the kth instance, hidden states at the kthinstance, measurement quality at the kth instance given a target typeand the plurality of measurements at the k−1^(th) time,z_(1, . . . k−1); p(z_(k)|T, x_(k), q_(k)) is a conditional probabilitydistribution function of the measurement at the kth instance given atarget type, the hidden states at the kth instance and the measurementquality at the kth instance; and p(x_(k), q_(k)|T,z_(1, 2, . . . , k−1)) is a conditional probability distributionfunction of the measurement at the kth instance and the measurementquality at the kth instance given a target type and the plurality ofmeasurements at the k−1^(th) time, z_(1, . . . k−1); and obtain anestimate of the target type from the first conditional probability. 25.The non-transitory computer readable medium of claim 24 whereinp(x_(k)|x_(k−1),T) is a multivariate Gaussian distribution given byp(x _(k) |x _(k−1) ,T)=N(x _(k);ƒ_(k)(x _(k−1) ,T),Q _(k)) where N(⋅; u,P) denotes a multivariate Gaussian with mean u and covariance P; andwherein p(x_(k)|T, z_(k)) is another multivariate Gaussian distribution,where p(x_(k)|x_(k−1),T) is a conditional probability distribution ofhidden states at instance k given hidden states at instance k−1 andtarget type T.
 26. The non-transitory computer readable medium of claim25 wherein transition between one hidden state at one instance and theone hidden state at another instance is given by a predetermined dynamicmodel (referred to as ƒ_(k)(x_(k−1), T)); wherein measurement quality isconsidered to be described by a Bernoulli distribution,${p\left( {\left. q_{k} \middle| q_{k - 1} \right.,T} \right)} = {{B\left( {{q_{k};q_{k - 1}},M_{k}} \right)} = \left\{ {\begin{matrix}{M_{k}q_{k - 1}} & {q_{k} = q_{k - 1}} \\{\left( {1 - M_{k}} \right)q_{k - 1}} & {q_{k} \neq q_{k - 1}}\end{matrix};} \right.}$ and wherein an expectation of p(x_(k)|T,z_(1, 2, . . . k−1)) is given byμ_(k,k−1) ^(x|q) ^(k) =∫ƒ_(k)(x _(k−1))N(x _(k);μ_(k−1,k−1) ^(x|q) ^(k),P _(k−1,k−1) ^(xx|q) ^(k) )dx _(k−1) and a covariance matrix is giveninductively byP _(k,k−1) ^(xx|q) ^(k) =Q _(k)+∫ƒ_(k)(x _(k−1))ƒ_(k) ^(T)(x _(k−1))N(x_(k);μ_(k−1,k−1) ^(x|q) ^(k) ,P _(k−1,k−1) ^(xx|q) ^(k) )dx_(k−1)−[μ_(k,k−1) ^(x|q) ^(k) ]^(T)μ_(k,k−1) ^(x|q) ^(k) .
 27. Thenon-transitory computer readable medium of claim 26 wherein measurementsare a predetermined function of target state and type (referred to ash_(k)(x_(k), T)) corrupted by independent noise; and wherein p(x_(k)|T,z_(1, 2, . . . k−1)) is a multivariate Gaussian distribution.
 28. Thenon-transitory computer readable medium of claim 27 whereinp(T|z_(1, 2, . . . , k)) is approximated by adaptive quadrature; andwherein${{p\left( T \middle| z_{1,2,\ldots,k} \right)} = {{p\left( {T,\left. z_{k} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}\text{/}{\sum\limits_{T}{p\left( {t^{\prime},\left. z_{k} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}}},{Where}$$\begin{matrix}{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( {\left. z_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}}} \\{= {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\sum\limits_{{q_{k} = 0},1}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k},q_{k}} \right)}{N\left( {{x_{k};\mu_{k,{k - 1}}^{{xx}|q_{k}}},P_{k,{k - 1}}^{{xx}|q_{k}}} \right)}{B\left( {{q_{k};\mu_{k,{k - 1}}^{q_{k}}},1} \right)}d\; x_{k}}}}}}\end{matrix}$ I = ∫p(z|x, T, q)N(x; μ^(x), P^(xx))d x$I = {{\frac{\sqrt{2}}{\left( {2\pi} \right)^{s\text{/}2}}{\int{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)} \right.,T,q} \right)}e^{{- v^{T}}v}d\; v}}} \approx {\sum\limits_{{i = 1},{\ldots|w|}}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T,q} \right)}w_{i}}}}$Where p(T, z_(k)|z_(1, 2, . . . k−1)) is a second conditionalprobability, p(z_(k)|T, x_(k)) is a third conditional probability, aFisher information is transformed by F=USV^(T), U & V are unitary, S isdiagonal, S_(ii)≥S_(jj) for i>j, ${s = {{\begin{matrix}\max \\i\end{matrix}\mspace{14mu} S_{ii}} \geq {ɛ\mspace{20mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)], w=Ûx,μ^(w) is an expectation of w at step i, P^(ww) is an expectation ofww^(T), Ũ is lower triangular and Ũ^(T)Ũ=P^(ww), a new variable v isdefined by${v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}},$a transformed Fisher information is {circumflex over(F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹, q₁={i:{circumflex over (F)}_(ii)≤α},q₂={i:{circumflex over (F)}_(ii)>α} where α is a constant, ∀i∈q₁ selectHermite order as |β{circumflex over (F)}_(ii)| where β is a constantyielding weights w_(i) and knots k_(i), ∀i∈q₂ select quadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\chi} \right\rceil$  where χ is aconstant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀].29. The non-transitory computer readable medium of claim 28 whereinproviding a first conditional probability comprises: inductivelyobtaining hidden state information at a kth stage from hidden stateinformation at a preceding (k−1)th stage; a obtaining a fourthconditional probability of a hidden state given a target type and anumber of measurements, p(x_(k), q_(k)|T, z_(1, 2, . . . , k−1)); andobtaining the first conditional probability, expressed inductively, as afunction of the fourth conditional probability.
 30. The non-transitorycomputer readable medium of claim 29 wherein in defining inductively thefirst conditional probability as$\mspace{76mu}{{{p\left( T \middle| z_{1,2,\ldots,k} \right)} = \frac{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}{\sum\limits_{\hat{T}}{p\left( {z_{k},\left. \hat{T} \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)}}},{{p\left( {z_{k},\left. T \middle| z_{1,2,\ldots,{k - 1}} \right.} \right)} = {{p\left( T \middle| z_{1,2,\ldots,{k - 1}} \right)}{\int{{p\left( {\left. z_{k} \middle| T \right.,x_{k}} \right)}{p\left( {\left. x_{k} \middle| T \right.,z_{1,2,\ldots,{k - 1}}} \right)}d\;{x_{k}.}}}}}}$31. The non-transitory computer readable medium of claim 30 wherein thethird and fourth conditional probabilities are represented bymultivariate Gaussian probability functions.
 32. The non-transitorycomputer readable medium of 31 wherein providing the first conditionalprobability distribution further comprises: using subspace partitioningto reduce dimensions of integration variable space of integralI=∫p(z|x,T,q)N(x;μ ^(x) ,P ^(xx))dx; transforming variables in theintegral in order to apply whitening; and applying differentapproximation methods to regions of more importance and regions of lessimportance, importance measured by Fisher information, in order toapproximate the integral by a sum,${\sum\limits_{{i = 1},{\ldots|w|}}{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{1}} + \mu^{w}} \right)} \right.,T} \right)}},{w_{i}.}$33. The non-transitory computer readable medium of claim 32 whereinusing subspace partitioning comprises: transforming the integrationvariable space by w=Ûx, Where a Fisher information is transformed byF=USV^(T), U & V are unitary, S is diagonal, S_(ii)≥S_(jj) for i>j,${s = {{\begin{matrix}\max \\i\end{matrix}\mspace{14mu} S_{ii}} \geq {ɛ\mspace{20mu}{for}\mspace{14mu}{some}\mspace{14mu}{small}\mspace{14mu} ɛ}}},$and an i-th column of U is denoted by U_(i) and Û=[U₁, . . . , U_(s)].34. The non-transitory computer readable medium of claim 33 whereintransforming variables in the integral in order to apply whiteningcomprises: transforming from w to a new variable v defined by$v = {\frac{1}{\sqrt{2}}{{\overset{\sim}{U}}^{- 1}\left( {w - \mu^{w}} \right)}}$in order to apply whitening.
 35. The non-transitory computer readablemedium of claim 34 wherein a transformed Fisher information is{circumflex over (F)}=[ÛŨ⁻¹]^(T)FÛŨ⁻¹, q₁={i:{circumflex over(F)}_(ii)≤α}, q₂={i:{circumflex over (F)}_(ii)>α} where α is a constant,for each i in q₁ select Hermite order as ┌β{circumflex over (F)}_(ii)┐where β is a constant yielding weights w_(i) and knots k_(i), for each iin q₂ select quadrature order as$\left\lceil \frac{{\hat{F}}_{ii}}{\chi} \right\rceil$  where χ is aconstant yielding weights w_(i) and knots k_(i), ⊗ denote thecolumn-wise Khatri-Rao product and Î_(i) a six column vector of onesw=w ₁ ⊗w ₂ ⊗ . . . ⊗w _(s)k=[Î ₀ k ₁ Î _(s−1)]⊗[Î ₁ k ₂ Î _(s−2)]⊗ . . . ⊗[Î _(s−1) k _(s) Î ₀],and wherein${\int{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}v} + \mu^{w}} \right)} \right.,T} \right)}e^{{- v^{T}}v}d\; v}} \approx {\sum\limits_{{i = 1},{\ldots|w|}}{{p\left( {\left. z \middle| {\hat{U}\left( {{\sqrt{2}\overset{\sim}{U}k_{i}} + \mu^{w}} \right)} \right.,T} \right)}{w_{i}.}}}$